10 research outputs found
Preferential attachment during the evolution of a potential energy landscape
It has previously been shown that the network of connected minima on a
potential energy landscape is scale-free, and that this reflects a power-law
distribution for the areas of the basins of attraction surrounding the minima.
Here, we set out to understand more about the physical origins of these
puzzling properties by examining how the potential energy landscape of a
13-atom cluster evolves with the range of the potential. In particular, on
decreasing the range of the potential the number of stationary points increases
and thus the landscape becomes rougher and the network gets larger. Thus, we
are able to follow the evolution of the potential energy landscape from one
with just a single minimum to a complex landscape with many minima and a
scale-free pattern of connections. We find that during this growth process, new
edges in the network of connected minima preferentially attach to more
highly-connected minima, thus leading to the scale-free character. Furthermore,
minima that appear when the range of the potential is shorter and the network
is larger have smaller basins of attraction. As there are many of these smaller
basins because the network grows exponentially, the observed growth process
thus also gives rise to a power-law distribution for the hyperareas of the
basins.Comment: 10 pages, 10 figure
Identifying "communities" within energy landscapes
Potential energy landscapes can be represented as a network of minima linked
by transition states. The community structure of such networks has been
obtained for a series of small Lennard-Jones clusters. This community structure
is compared to the concept of funnels in the potential energy landscape. Two
existing algorithms have been used to find community structure, one involving
removing edges with high betweenness, the other involving optimization of the
modularity. The definition of the modularity has been refined, making it more
appropriate for networks such as these where multiple edges and
self-connections are not included. The optimization algorithm has also been
improved, using Monte Carlo methods with simulated annealing and basin hopping,
both often used successfully in other optimization problems. In addition to the
small clusters, two examples with known heterogeneous landscapes, LJ_13 with
one labelled atom and LJ_38, were studied with this approach. The network
methods found communities that are comparable to those expected from landscape
analyses. This is particularly interesting since the network model does not
take any barrier heights or energies of minima into account. For comparison,
the network associated with a two-dimensional hexagonal lattice is also studied
and is found to have high modularity, thus raising some questions about the
interpretation of the community structure associated with such partitions.Comment: 13 pages, 11 figure
Power-law distributions for the areas of the basins of attraction on a potential energy landscape
Energy landscape approaches have become increasingly popular for analysing a
wide variety of chemical physics phenomena. Basic to many of these applications
has been the inherent structure mapping, which divides up the potential energy
landscape into basins of attraction surrounding the minima. Here, we probe the
nature of this division by introducing a method to compute the basin area
distribution and applying it to some archetypal supercooled liquids. We find
that this probability distribution is a power law over a large number of
decades with the lower-energy minima having larger basins of attraction.
Interestingly, the exponent for this power law is approximately the same as
that for a high-dimensional Apollonian packing, providing further support for
the suggestion that there is a strong analogy between the way the energy
landscape is divided into basins, and the way that space is packed in
self-similar, space-filling hypersphere packings, such as the Apollonian
packing. These results suggest that the basins of attraction provide a
fractal-like tiling of the energy landscape, and that a scale-free pattern of
connections between the minima is a general property of energy landscapes.Comment: 4 pages, 3 figure
Characterizing the network topology of the energy landscapes of atomic clusters
By dividing potential energy landscapes into basins of attractions
surrounding minima and linking those basins that are connected by transition
state valleys, a network description of energy landscapes naturally arises.
These networks are characterized in detail for a series of small Lennard-Jones
clusters and show behaviour characteristic of small-world and scale-free
networks. However, unlike many such networks, this topology cannot reflect the
rules governing the dynamics of network growth, because they are static spatial
networks. Instead, the heterogeneity in the networks stems from differences in
the potential energy of the minima, and hence the hyperareas of their
associated basins of attraction. The low-energy minima with large basins of
attraction act as hubs in the network.Comparisons to randomized networks with
the same degree distribution reveals structuring in the networks that reflects
their spatial embedding.Comment: 14 pages, 11 figure
Exploring the origins of the power-law properties of energy landscapes: An egg-box model
Multidimensional potential energy landscapes (PELs) have a Gaussian
distribution for the energies of the minima, but at the same time the
distribution of the hyperareas for the basins of attraction surrounding the
minima follows a power-law. To explore how both these features can
simultaneously be true, we introduce an ``egg-box'' model. In these model
landscapes, the Gaussian energy distribution is used as a starting point and we
examine whether a power-law basin area distribution can arise as a natural
consequence through the swallowing up of higher-energy minima by larger
low-energy basins when the variance of this Gaussian is increased sufficiently.
Although the basin area distribution is substantially broadened by this
process,it is insufficient to generate power-laws, highlighting the role played
by the inhomogeneous distribution of basins in configuration space for actual
PELs.Comment: 7 pages, 8 figure
A self-consistent approach to measure preferential attachment in networks and its application to an inherent structure network
Preferential attachment is one possible way to obtain a scale-free network.
We develop a self-consistent method to determine whether preferential
attachment occurs during the growth of a network, and to extract the
preferential attachment rule using time-dependent data. Model networks are
grown with known preferential attachment rules to test the method, which is
seen to be robust. The method is then applied to a scale-free inherent
structure network, which represents the connections between minima via
transition states on a potential energy landscape. Even though this network is
static, we can examine the growth of the network as a function of a threshold
energy (rather than time), where only those transition states with energies
lower than the threshold energy contribute to the network.For these networks we
are able to detect the presence of preferential attachment, and this helps to
explain the ubiquity of funnels on energy landscapes. However, the scale-free
degree distribution shows some differences from that of a model network grown
using the obtained preferential attachment rules, implying that other factors
are also important in the growth process.Comment: 8 pages, 8 figure
Self-similar disk packings as model spatial scale-free networks
The network of contacts in space-filling disk packings, such as the
Apollonian packing, are examined. These networks provide an interesting example
of spatial scale-free networks, where the topology reflects the broad
distribution of disk areas. A wide variety of topological and spatial
properties of these systems are characterized. Their potential as models for
networks of connected minima on energy landscapes is discussed.Comment: 13 pages, 12 figures; some bugs fixed and further discussion of
higher-dimensional packing